INTERNATIONAL GONGRESS $\begin{array}{rl}& \text{OF MATHEMATIGIANS}\\ & 2022\text{JULY}6-14\end{array}$$\begin{array}{r}\text{ OF MATHEMATIGIANS }\\ 2022\text{ JULY }6−14\end{array}${:[" OF MATHEMATIGIANS "],[2022" JULY "6-14]:}\begin{aligned} & \text { OF MATHEMATIGIANS } \\ & 2022 \text { JULY } 6-14\end{aligned}$\begin{array}{rl}& \text{ OF MATHEMATIGIANS }\\ & 2022\text{ JULY }6−14\end{array}$

Dmitry Beliaev

Mathematical Institute

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford OX2 6GG, UK

Email: belyaev@maths.ox.ac.uk
Stanislav Smirnov

Section de mathématiques

Université de Genève rue du Conseil-Général 7-9

1205 Genève, Switzerland

Email: stanislav.smirnov@unige.ch

2020 Mathematics Subject Classification: 00B25

ISBN 978-3-98547-058-7, eISBN 978-3-98547-558-2, DOI 10.4171/ICM2O22

ISBN 978-3-98547-059-4, eISBN 978-3-98547-559-9, DOI 10.4171/ICM2022-1

Volume 2. Plenary Lectures

ISBN 978-3-98547-060-0, eISBN 978-3-98547-560-5, DOI 10.4171/ICM2022-2

$\to$$→$rarr\rightarrow$→$ Volume 3. Sections 1-4

ISBN 978-3-98547-061-7, eISBN 978-3-98547-561-2, DOI 10.4171/ICM2022-3

Volume 4. Sections 5-8

ISBN 978-3-98547-062-4, eISBN 978-3-98547-562-9, DOI 10.4171/ICM2022-4

Volume 5. Sections 9-11

ISBN 978-3-98547-063-1, eISBN 978-3-98547-563-6, DOI 10.4171/ICM2022-5

Volume 6. Sections 12-14

ISBN 978-3-98547-064-8, eISBN 978-3-98547-564-3, DOI 10.4171/ICM2022-6

Volume 7. Sections 15-20

ISBN 978-3-98547-065-5, eISBN 978-3-98547-565-0, DOI 10.4171/ICM2022-7

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Foreword ..... V
International Congresses of Mathematicians ..... 1
Fields medalists and IMU prize winners ..... 3
Opening greetings by the IMU President ..... 5
Closing remarks by the IMU President ..... 9
Status report for the IMU ..... 11
Photographs ..... 21
THE WORK OF THE FIELDS MEDALISTS AND THE IMU PRIZE WINNERS
Martin Hairer, The work of Hugo Duminil-Copin ..... 26
Gil Kalai, The work of June Huh ..... 50
Kannan Soundararajan, The work of James Maynard ..... 66
Henry Cohn, The work of Maryna Viazovska ..... 82
Ran Raz, The work of Mark Braverman ..... 106
Henri Darmon, The work of Barry Mazur ..... 118
Rupert L. Frank, The work of Elliott Lieb ..... 142
Tadashi Tokieda, Nikolai Andreev and the art of mathematical animation and model-
building ..... 160

Hugo Duminil-Copin, 100 years of the (critical) Ising model on the hypercubic
lattice ..... 164
June Huh, Combinatorics and Hodge theory ..... 212
James Maynard, Counting primes ..... 240
Maryna Viazovska, On discrete Fourier uniqueness sets in Euclidean space ..... 270
Mark Braverman, Communication and information complexity ..... 284
Nikolai Andreev, Popularization of math: sketches of Russian projects and traditions ..... 322
Marie-France Vignéras, Representations of $p$$p$pp$p$-adic groups over commutative rings ..... 332
POPULAR SCIENTIFIC EXPOSITIONS
Andrei Okounkov, The Ising model in our dimension and our times ..... 376
Andrei Okounkov, Combinatorial geometry takes the lead ..... 414
Andrei Okounkov, Rhymes in primes ..... 460
Andrei Okounkov, The magic of 8 and 24 ..... 492
SUMMARIES OF PRIZE WINNERS' WORK
Allyn Jackson, 2022 Abacus Medal: Mark Braverman ..... 548
Allyn Jackson, 2022 Chern Medal: Barry Mazur ..... 554
Allyn Jackson, 2022 Gauss Prize: Elliott H. Lieb ..... 560
Allyn Jackson, 2022 Leelavati Prize: Nikolai Andreev ..... 566
List of contributors ..... 571

Kevin Buzzard, What is the point of computers? A question for pure mathematicians ..... 578
Frank Calegari, Reciprocity in the Langlands program since Fermat's Last Theorem ..... 610
Frans Pretorius, A survey of gravitational waves ..... 652
PLENARY LECTURES
Mladen Bestvina, Groups acting on hyperbolic spaces-a survey ..... 678
Bhargav Bhatt, Algebraic geometry in mixed characteristic ..... 712
Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella,
Dynamics of dilute gases: a statistical approach ..... 750
Alexander Braverman, David Kazhdan, Automorphic functions on moduli spaces of
bundles on curves over local fields: a survey ..... 796
Tobias Holck Colding, Evolution of form and shape ..... 826
Camillo De Lellis, The regularity theory for the area functional (in geometric mea-
sure theory) ..... 872
Weinan E, A mathematical perspective of machine learning ..... 914
Craig Gentry, Homomorphic encryption: a mathematical survey ..... 956
Alice Guionnet, Rare events in random matrix theory ..... 1008
Larry Guth, Decoupling estimates in Fourier analysis ..... 1054
Svetlana Jitomirskaya, One-dimensional quasiperiodic operators: global theory, dual-
ity, and sharp analysis of small denominators ..... 1090
Igor Krichever, Abelian pole systems and Riemann-Schottky-type problems . . ..... 1122
Alexander Kuznetsov, Semiorthogonal decompositions in families ..... 1154
Scott Sheffield, What is a random surface? ..... 1202
Kannan Soundararajan, The distribution of values of zeta and L-functions ..... 1260
Catharina Stroppel, Categorification: tangle invariants and TQFTs ..... 1312
Michel Van den Bergh, Noncommutative crepant resolutions, an overview ..... 1354
Avi Wigderson, Interactions of computational complexity theory and mathematics ..... 1392
List of contributors ..... 1433

Gal Binyamini, Dmitry Novikov, Tameness in geometry and arithmetic: beyond
o-minimality ..... 1440
Natasha Dobrinen, Ramsey theory of homogeneous structures: current trends and
open problems ..... 1462
Andrew S. Marks, Measurable graph combinatorics ..... 1488
Keita Yokoyama, The Paris-Harrington principle and second-order arithmetic-
bridging the finite and infinite Ramsey theorem ..... 1504

Pierre-Emmanuel Caprace, George A. Willis, A totally disconnected invitation to locally compact groups 1554

Neena Gupta, The Zariski cancellation problem and related problems in affine algebraic geometry

Syu Kato, The formal model of semi-infinite flag manifolds ..................... 1600

Michael J. Larsen, Character estimates for finite simple groups and applications . . 1624

Amnon Neeman, Finite approximations as a tool for studying triangulated categories 1636

Irena Peeva, Syzygies over a polynomial ring ....................................... 1660

Joseph H. Silverman, Survey lecture on arithmetic dynamics

1682

Ana Caraiani, The cohomology of Shimura varieties with torsion coefficients . . . 1744

Samit Dasgupta, Mahesh Kakde, On the Brumer-Stark conjecture and refinements 1768

Alexander Gamburd, Arithmetic and dynamics on varieties of Markoff type ...... 1800

Philipp Habegger, The number of rational points on a curve of genus at least two . 1838

Atsushi Ichino, Theta lifting and Langlands functoriality ........................ 1870

Dimitris Koukoulopoulos, Rational approximations of irrational numbers ........ 1894

David Loeffler, Sarah Livia Zerbes, Euler systems and the Bloch-Kato conjecture for
automorphic Galois representations . ................................................................. 1918

Lillian B. Pierce, Counting problems: class groups, primes, and number fields . . . 1940

Sug Woo Shin, Points on Shimura varieties modulo primes ....................... 1966

Ye Tian, The congruent number problem and elliptic curves .......................... 1990

Xinwen Zhu, Arithmetic and geometric Langlands program ....................... 2012

Mina Aganagic, Homological knot invariants from mirror symmetry ..... 2108
Aravind Asok, Jean Fasel, Vector bundles on algebraic varieties ..... 2146
Arend Bayer, Emanuele Macrì, The unreasonable effectiveness of wall-crossing in
algebraic geometry ..... 2172
Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich, Counting lattice
points in moduli spaces of quadratic differentials ..... 2196
Alexander I. Efimov, K-theory of large categories ..... 2212
Tamás Hausel, Enhanced mirror symmetry for Langlands dual Hitchin systems ..... 2228
Bruno Klingler, Hodge theory, between algebraicity and transcendence ..... 2250
Chi Li, Canonical Kähler metrics and stability of algebraic varieties ..... 2286
Aaron Pixton, The double ramification cycle formula ..... 2312
Yuri Prokhorov, Effective results in the three-dimensional minimal model program ..... 2324
Olivier Wittenberg, Some aspects of rational points and rational curves ..... 2346
List of contributors ..... 2369

Bruce Kleiner, Developments in 3D Ricci flow since Perelman ..... 2376
Richard Evan Schwartz, Survey lecture on billiards ..... 2392
5. GEOMETRY
Richard H. Bamler, Some recent developments in Ricci flow ..... 2432
Robert J. Berman, Emergent complex geometry ..... 2456
Danny Calegari, Sausages ..... 2484
Kai Cieliebak, Lagrange multiplier functionals and their applications in symplectic
geometry and string topology ..... 2504
Penka Georgieva, Real Gromov-Witten theory ..... 2530
Hiroshi Iritani, Gamma classes and quantum cohomology ..... 2552
Gang Liu, Kähler manifolds with curvature bounded below ..... 2576
Kathryn Mann, Groups acting at infinity ..... 2594
Mark McLean, Floer cohomology, singularities, and birational geometry ..... 2616
Iskander A. Taimanov, Surfaces via spinors and soliton equations ..... 2638
Lu Wang, Entropy in mean curvature flow ..... 2656
Robert J. Young, Composing and decomposing surfaces and functions ..... 2678
Xin Zhou, Mean curvature and variational theory ..... 2696
Xiaohua Zhu, Kähler-Ricci flow on Fano manifolds ..... 2718
6. TOPOLOGY
Jennifer Hom, Homology cobordism, knot concordance, and Heegaard Floer homol-
ogy ..... 2740
Daniel C. Isaksen, Guozhen Wang, Zhouli Xu, Stable homotopy groups of spheres
and motivic homotopy theory ..... 2768
Yi Liu, Surface automorphisms and finite covers ..... 2792
Roman Mikhailov, Homotopy patterns in group theory ..... 2806
Thomas Nikolaus, Frobenius homomorphisms in higher algebra ..... 2826
Oscar Randal-Williams, Diffeomorphisms of discs ..... 2856
Jacob Rasmussen, Floer homology of 3-manifolds with torus boundary ..... 2880
Nathalie Wahl, Homological stability: a tool for computations ..... 2904
7. LIE THEORY AND GENERALIZATIONS
Evgeny Feigin, PBW degenerations, quiver Grassmannians, and toric varieties ..... 2930
Tasho Kaletha, Representations of reductive groups over local fields . . ..... 2948
Joel Kamnitzer, Perfect bases in representation theory: three mountains and their
springs ..... 2976
Yiannis Sakellaridis, Spherical varieties, functoriality, and quantization ..... 2998
Peng Shan, Categorification and applications ..... 3038
Binyong Sun, Chen-Bo Zhu, Theta correspondence and the orbit method ..... 3062
Weiqiang Wang, Quantum symmetric pairs ..... 3080

Keith Ball, Convex geometry and its connections to harmonic analysis, functional analysis and probability theory ..... 3104

Benoît Collins, Moment methods on compact groups: Weingarten calculus and its
applications ..... 3142
Mikael de la Salle, Analysis on simple Lie groups and lattices ..... 3166
Xiumin Du, Weighted Fourier extension estimates and applications ..... 3190
Cyril Houdayer, Noncommutative ergodic theory of higher rank lattices ..... 3202
Malabika Pramanik, On some properties of sparse sets: a survey ..... 3224
Gideon Schechtman, The number of closed ideals in the algebra of bounded operators
on Lebesgue spaces ..... 3250
Pablo Shmerkin, Slices and distances: on two problems of Furstenberg and Falconer ..... 3266
Konstantin Tikhomirov, Quantitative invertibility of non-Hermitian random matrices ..... 3292
Stuart White, Abstract classification theorems for amenable ${\mathrm{C}}^{\ast }$${\mathrm{C}}^{∗}$C^(**)\mathrm{C}^{*}${\mathrm{C}}^{∗}$-algebras ..... 3314
Tianyi Zheng, Asymptotic behaviors of random walks on countable groups ..... 3340
List of contributors ..... 3367
VOLUME 5
9. DYNAMICS
Miklós Abért, On a curious problem and what it lead to ..... 3374
Aaron Brown, Lattice subgroups acting on manifolds ..... 3388
Jon Chaika, Barak Weiss, The horocycle flow on the moduli space of translation
surfaces ..... 3412
Mark F. Demers, Topological entropy and pressure for finite-horizon Sinai billiards ..... 3432
Romain Dujardin, Geometric methods in holomorphic dynamics ..... 3460
David Fisher, Rigidity, lattices, and invariant measures beyond homogeneous dynam-
ics ..... 3484
Mariusz Lemańczyk, Furstenberg disjointness, Ratner properties, and Sarnak's con-
jecture ..... 3508
Amir Mohammadi, Finitary analysis in homogeneous spaces ..... 3530
Michela Procesi, Stability and recursive solutions in Hamiltonian PDEs ... ..... 3552
Corinna Ulcigrai, Dynamics and "arithmetics" of higher genus surface flows ..... 3576
Péter P. Varjú, Self-similar sets and measures on the line ..... 3610

Tristan Buckmaster, Theodore D. Drivas, Steve Shkoller, Vlad Vicol, Formation and
development of singularities for the compressible Euler equations ..... 3636
Pierre Cardaliaguet, François Delarue, Selected topics in mean field games ..... 3660
Semyon Dyatlov, Macroscopic limits of chaotic eigenfunctions ..... 3704
Rita Ferreira, Irene Fonseca, Raghavendra Venkatraman, Variational homogeniza-
tion: old and new ..... 3724
Rupert L. Frank, Lieb-Thirring inequalities and other functional inequalities for
orthonormal systems ..... 3756
Alexandru D. Ionescu, Hao Jia, On the nonlinear stability of shear flows and vortices ..... 3776
Mathieu Lewin, Mean-field limits for quantum systems and nonlinear Gibbs mea-
sures ..... 3800
Kenji Nakanishi, Global dynamics around and away from solitons ..... 3822
Alexander I. Nazarov, Variety of fractional Laplacians ..... 3842
Galina Perelman, Formation of singularities in nonlinear dispersive PDEs ..... 3854
Gabriella Tarantello, On the asymptotics for minimizers of Donaldson functional in
Teichmüller theory ..... 3880
Dongyi Wei, Zhifei Zhang, Hydrodynamic stability at high Reynolds number ..... 3902
11. MATHEMATICAL PHYSICS - SPECIAL LECTURE
Peter Hintz, Gustav Holzegel, Recent progress in general relativity ..... 3924
11. MATHEMATICAL PHYSICS
Roland Bauerschmidt, Tyler Helmuth, Spin systems with hyperbolic symmetry:
a survey ..... 3986
Federico Bonetto, Eric Carlen, Michael Loss, The Kac model: variations on a theme ..... 4010
Søren Fournais, Jan Philip Solovej, On the energy of dilute Bose gases ..... 4026
Alessandro Giuliani, Scaling limits and universality of Ising and dimer models . . ..... 4040
Matthew B. Hastings, Gapped quantum systems: from higher-dimensional Lieb-
Schultz-Mattis to the quantum Hall effect ..... 4074
Karol Kajetan Kozlowski, Bootstrap approach to 1+1-dimensional integrable quan-
tum field theories: the case of the sinh-Gordon model ..... 4096
Jonathan Luk, Singularities in general relativity ..... 4120
Yoshiko Ogata, Classification of gapped ground state phases in quantum spin sys-
tems ..... 4142
List of contributors ..... 4163
VOLUME 6
12. PROBABILITY - SPECIAL LECTURE
Elchanan Mossel, Combinatorial statistics and the sciences ..... 4170
12. PROBABILITY
Jinho Baik, KPZ limit theorems ..... 4190
Jian Ding, Julien Dubédat, Ewain Gwynne, Introduction to the Liouville quantum
gravity metric ..... 4212
Ronen Eldan, Analysis of high-dimensional distributions using pathwise methods ..... 4246
Alison Etheridge, Natural selection in spatially structured populations ..... 4272
Tadahisa Funaki, Hydrodynamic limit and stochastic PDEs related to interface
motion ..... 4302
Patrícia Gonçalves, On the universality from interacting particle systems ..... 4326
Hubert Lacoin, Mixing time and cutoff for one-dimensional particle systems ..... 4350
Dmitry Panchenko, Ultrametricity in spin glasses ..... 4376
Kavita Ramanan, Interacting stochastic processes on sparse random graphs ..... 4394
Daniel Remenik, Integrable fluctuations in the KPZ universality class ..... 4426
Laurent Saloff-Coste, Heat kernel estimates on Harnack manifolds and beyond ..... 4452
13. COMBINATORICS - SPECIAL LECTURE
Melanie Matchett Wood, Probability theory for random groups arising in number
theory

4476
13. COMBINATORICS
Federico Ardila-Mantilla, The geometry of geometries: matroid theory, old and new ..... 4510
Julia Böttcher, Graph and hypergraph packing ..... 4542
Ehud Friedgut, KKL's influence on me ..... 4568
Allen Knutson, Schubert calculus and quiver varieties ..... 4582
Sergey Norin, Recent progress towards Hadwiger's conjecture ..... 4606
Isabella Novik, Face numbers: the upper bound side of the story ..... 4622
Mathias Schacht, Restricted problems in extremal combinatorics ..... 4646
Alex Scott, Graphs of large chromatic number ..... 4660
Asaf Shapira, Local-vs-global combinatorics ..... 4682
Lauren K. Williams, The positive Grassmannian, the amplituhedron, and cluster alge-
bras ..... 4710
14. MATHEMATICS OF COMPUTER SCIENCE - SPECIAL LECTURES
Cynthia Dwork, Differential privacy: getting more for less ..... 4740
Aayush Jain, Huijia Lin, Amit Sahai, Indistinguishability obfuscation ... ..... 4762
David Silver, Andre Barreto, Simulation-based search control ..... 4800
Bernd Sturmfels, Beyond linear algebra ..... 4820
14. MATHEMATICS OF COMPUTER SCIENCE
Roy Gotlib, Tali Kaufman, Nowhere to go but high: a perspective on high-dimensional
expanders ..... 4842
Jelani Nelson, Forty years of frequent items ..... 4872
Oded Regev, Some questions related to the reverse Minkowski theorem ..... 4898
Muli (Shmuel) Safra, Mathematics of computation through the lens of linear equa-
tions and lattices ..... 4914
Ola Svensson, Polyhedral techniques in combinatorial optimization: matchings and
tours ..... 4970
Thomas Vidick, MIP* $=$$=$==$=$ RE: a negative resolution to Connes' embedding problem
and Tsirelson's problem ..... 4996
List of contributors ..... 5027

Gang Bao, Mathematical analysis and numerical methods for inverse scattering problems
Marsha J. Berger, Randall J. LeVeque, Towards adaptive simulations of dispersive
tsunami propagation from an asteroid impact ..... 5056
Jan S. Hesthaven, Cecilia Pagliantini, Nicolò Ripamonti, Structure-preserving model
order reduction of Hamiltonian systems ..... 5072
Nicholas J. Higham, Numerical stability of algorithms at extreme scale and low pre-
cisions ..... 5098
Gitta Kutyniok, The mathematics of artificial intelligence ..... 5118
Rachel Ward, Stochastic gradient descent: where optimization meets machine learn-
ing ..... 5140
Lexing Ying, Solving inverse problems with deep learning ..... 5154
16. CONTROL THEORY AND OPTIMIZATION - SPECIAL LECTURE
Nikhil Bansal, Discrepancy theory and related algorithms ..... 5178
16. CONTROL THEORY AND OPTIMIZATION
Regina S. Burachik, Enlargements: a bridge between maximal monotonicity and con-
vexity ..... 5212
Martin Burger, Nonlinear eigenvalue problems for seminorms and applications ..... 5234
Coralia Cartis, Nicholas I. M. Gould, Philippe L. Toint, The evaluation complexity
of finding high-order minimizers of nonconvex optimization ..... 5256
Yu-Hong Dai, An overview of nonlinear optimization ..... 5290
Qi Lü, Control theory of stochastic distributed parameter systems: recent progress
and open problems ..... 5314
Asuman Ozdaglar, Muhammed O. Sayin, Kaiqing Zhang, Independent learning in
stochastic games ..... 5340
Marius Tucsnak, Reachable states for infinite-dimensional linear systems: old and
new ..... 5374
17. STATISTICS AND DATA ANALYSIS
Francis Bach, Lénaïc Chizat, Gradient descent on infinitely wide neural networks:
global convergence and generalization ..... 5398
Bin Dong, On mathematical modeling in image reconstruction and beyond . . ..... 5420
Stefanie Jegelka, Theory of graph neural networks: representation and learning ..... 5450
Oleg V. Lepski, Theory of adaptive estimation ..... 5478
Gábor Lugosi, Mean estimation in high dimension ..... 5500
Richard Nickl, Gabriel P. Paternain, On some information-theoretic aspects of non-
linear statistical inverse problems ..... 5516
Bernhard Schölkopf, Julius von Kügelgen, From statistical to causal learning ..... 5540
Cun-Hui Zhang, Second- and higher-order Gaussian anticoncentration inequalities
and error bounds in Slepian's comparison theorem ..... 5594
18. STOCHASTIC AND DIFFERENTIAL MODELLING
Jacob Bedrossian, Alex Blumenthal, Sam Punshon-Smith, Lower bounds on the Lya-
punov exponents of stochastic differential equations ..... 5618
Nicolas Champagnat, Sylvie Méléard, Viet Chi Tran, Multiscale eco-evolutionary
models: from individuals to populations ..... 5656
Hyeonbae Kang, Quantitative analysis of field concentration in presence of closely
located inclusions of high contrast ..... 5680
19. MATHEMATICAL EDUCATION AND POPULARIZATION OF MATHEMATICS
Clara I. Grima, The hug of the scutoid ..... 5702
Anna Sfard, The long way from mathematics to mathematics education: how edu-
cational research may change one's vision of mathematics and of its learning and
teaching ..... 5716
20. HISTORY OF MATHEMATICS
June Barrow-Green, George Birkhoff's forgotten manuscript and his programme for
dynamics ..... 5748
Annette Imhausen, Some uses and associations of mathematics, as seen from a distant
historical perspective ..... 5772
Krishnamurthi Ramasubramanian, The history and historiography of the discovery
of calculus in India ..... 5784
List of contributors ..... 5813

Primary 03C64; Secondary 11U09, 32B20, 14G05

o-minimality, point counting, Pila-Wilkie theorem, André-Oort theorem, Yomdin-Gromov parameterization

The theory of o-minimal structures was introduced by van den Dries as an attempt to provide a framework of tame topology in the spirit of Grothendieck's "Esquisse d'un Programme" [42]. We refer the reader to this book for a general introduction to the subject and its history. For us, an o-minimal structure will always be an expansion of the ordered real field ${\mathbb{R}}_{\text{alg}}:=\{\mathbb{R},+,\cdot ,<\}$${\mathbb{R}}_{\text{alg }}:=\{\mathbb{R},+,â‹\dots ,<\}$R_("alg "):={R,+,*, < }\mathbb{R}_{\text {alg }}:=\{\mathbb{R},+, \cdot,<\}${\mathbb{R}}_{\text{alg }}:=\{\mathbb{R},+,â‹\dots ,<\}$. Briefly, such an expansion is o-minimal if all definable subsets of $\mathbb{R}$$\mathbb{R}$R\mathbb{R}$\mathbb{R}$ consist of finite unions of points and intervals.

Despite their apparent simplicity, it turns out that the axioms of o-minimality provide a broad framework of tame topology. In particular, one has good notions of dimension, smooth stratification, triangulation, and cell-decomposition for every definable set in an o-minimal structure. On the other hand, several natural and important structures turn out to be o-minimal. A few examples of particular importance for us in the present paper are ${\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}},{\mathbb{R}}_{\mathrm{a}\mathrm{n}},{\mathbb{R}}_{\mathrm{a}\mathrm{n},\mathrm{e}\mathrm{x}\mathrm{p}}$${\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}},{\mathbb{R}}_{\mathrm{a}\mathrm{n}},{\mathbb{R}}_{\mathrm{a}\mathrm{n},\mathrm{e}\mathrm{x}\mathrm{p}}$R_(alg),R_(an),R_(an,exp)\mathbb{R}_{\mathrm{alg}}, \mathbb{R}_{\mathrm{an}}, \mathbb{R}_{\mathrm{an}, \mathrm{exp}}${\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}},{\mathbb{R}}_{\mathrm{a}\mathrm{n}},{\mathbb{R}}_{\mathrm{a}\mathrm{n},\mathrm{e}\mathrm{x}\mathrm{p}}$, and ${\mathbb{R}}_{\text{Pfaff.}}$${\mathbb{R}}_{\text{Pfaff. }}$R_("Pfaff. ")\mathbb{R}_{\text {Pfaff. }}${\mathbb{R}}_{\text{Pfaff. }}$. We will say a bit more on these in later sections.

In [37], Pila and Wilkie discovered a "counting theorem" that would later find deep applications in arithmetic geometry. The theorem concerns the asymptotic density of rational (or algebraic) points in a definable set-as a function of height. We introduce this first, to motivate a broader discussion of the connection between tame geometry and arithmetic.

For $x\in \mathbb{Q}$$x∈\mathbb{Q}$x inQx \in \mathbb{Q}$x∈\mathbb{Q}$, we denote by $H(x)$$H(x)$H(x)H(x)$H(x)$ the standard height of $x$$x$xx$x$. For a vector $x\in {\mathbb{Q}}^n$$x∈{\mathbb{Q}}^n$x inQ^(n)x \in \mathbb{Q}^{n}$x∈{\mathbb{Q}}^n$, we denote by $H(x)$$H(x)$H(x)H(x)$H(x)$ the maximum among the heights of the coordinates of $x$$x$xx$x$. For a set $A\subset {\mathbb{R}}^n$$A⊂{\mathbb{R}}^n$A subR^(n)A \subset \mathbb{R}^{n}$A⊂{\mathbb{R}}^n$, we denote the set of $\mathbb{Q}$$\mathbb{Q}$Q\mathbb{Q}$\mathbb{Q}$-points of $A$$A$AA$A$ by $A(\mathbb{Q})=A\cap {\mathbb{Q}}^n$$A(\mathbb{Q})=Aâˆ\copyright {\mathbb{Q}}^n$A(Q)=A nnQ^(n)A(\mathbb{Q})=A \cap \mathbb{Q}^{n}$A(\mathbb{Q})=Aâˆ\copyright {\mathbb{Q}}^n$ and denote

For a set $A\subset {\mathbb{R}}^n$$A⊂{\mathbb{R}}^n$A subR^(n)A \subset \mathbb{R}^{n}$A⊂{\mathbb{R}}^n$, we define the algebraic part $A^{\text{alg}}$$A^{\text{alg }}$A^("alg ")A^{\text {alg }}$A^{\text{alg }}$ of $A$$A$AA$A$ to be the union of all connected semialgebraic subsets of $A$$A$AA$A$ of positive dimension. We define the transcendental part $A^{\text{trans}}$$A^{\text{trans }}$A^("trans ")A^{\text {trans }}$A^{\text{trans }}$ of $A$$A$AA$A$ to be $A\mathrm{\setminus }{A}^{\text{alg}}$$A\mathrm{∖}{A}^{\text{alg }}$A\\A^("alg ")A \backslash A^{\text {alg }}$A\mathrm{∖}{A}^{\text{alg }}$.

Theorem 1 (Pila and Wilkie [37]). Let $A\subset {\mathbb{R}}^m$$A⊂{\mathbb{R}}^m$A subR^(m)A \subset \mathbb{R}^{m}$A⊂{\mathbb{R}}^m$ be a set definable in an o-minimal structure. Then for every $ϵ>0$$\mathrm{Ï\mu }>0$epsilon > 0\epsilon>0$\mathrm{Ï\mu }>0$ there exists a constant $C(A,ϵ)$$C(A,\mathrm{Ï\mu })$C(A,epsilon)C(A, \epsilon)$C(A,\mathrm{Ï\mu })$ such that for every $H⩾1$$Hâ\copyright ¾1$H >= 1H \geqslant 1$Hâ\copyright ¾1$,

The use of transcendental (as opposed to algebraic) methods in the study of arithmetic questions has a long history. A common theme in these methods, running through the work of Schneider, Lang, Baker, Masser, and Wüstholz to name a few, is the use of auxiliary polynomials. We refer to [28] for a broad treatment of this subject.

The usefulness of polynomials in this context stems from their dual algebraic/analytic role. Suppose one is interested in the set $A(\mathbb{Q},H)$$A(\mathbb{Q},H)$A(Q,H)A(\mathbb{Q}, H)$A(\mathbb{Q},H)$ for some analytic set $A$$A$AA$A$. On the one hand, if a polynomial $P$$P$PP$P$, say, with integer coefficients, is evaluated at $x\in A(\mathbb{Q},H)$$x∈A(\mathbb{Q},H)$x in A(Q,H)x \in A(\mathbb{Q}, H)$x∈A(\mathbb{Q},H)$ then $P(x)$$P(x)$P(x)P(x)$P(x)$ is again rational, and one can estimate its height in terms of $H$$H$HH$H$ and the height of $P$$P$PP$P$. On
the other hand, polynomials are extremely well-behaved analytic functions, and a variety of analytic methods may be used to prove upper bounds on the restriction of $P$$P$PP$P$ to an analytic set $A$$A$AA$A$ assuming it is appropriately constructed (say to vanish to high order at some points of $A)$$A)$A)A)$A)$. One concludes from such an argument that $P$$P$PP$P$ must vanish at every point in $A(\mathbb{Q},H)$$A(\mathbb{Q},H)$A(Q,H)A(\mathbb{Q}, H)$A(\mathbb{Q},H)$, for otherwise the height bound would contradict the upper bound.

The proof of the Pila-Wilkie counting theorem follows this classical line. However, it is fairly unique in the realm of transcendence methods in that the degrees of the auxiliary polynomials $P$$P$PP$P$ are independent of the height, depending in fact only on $\epsilon$$\mathrm{Î\mu }$epsi\varepsilon$\mathrm{Î\mu }$. It is this unusual feature that makes it possible to prove the Pila-Wilkie theorem in the vast generality of o-minimal structures: polynomials of a given degree form a definable family, and the general machinery of o-minimality gives various finiteness statements uniformly for all such polynomials.

By contrast with the Pila-Wilkie theorem, most transcendence methods require the degrees of the auxiliary polynomials to depend on the height $H$$H$HH$H$ of the points being considered-sometimes logarithmically and in some cases, such as the Schneider-Lang theorem, even linearly. A famous conjecture that seems to fall within this category is due to Wilkie.

Conjecture 2 (Wilkie [37]). Let $A\subset {\mathbb{R}}^m$$A⊂{\mathbb{R}}^m$A subR^(m)A \subset \mathbb{R}^{m}$A⊂{\mathbb{R}}^m$ be a set definable in ${\mathbb{R}}_{\exp }$${\mathbb{R}}_{\exp }$R_(exp)\mathbb{R}_{\exp }${\mathbb{R}}_{\exp }$. Then there exist constants $C(A),\kappa (A)$$C(A),\mathrm{κ}(A)$C(A),kappa(A)C(A), \kappa(A)$C(A),\mathrm{κ}(A)$ such that for all $H⩾3$$Hâ\copyright ¾3$H >= 3H \geqslant 3$Hâ\copyright ¾3$,

The conclusion of the Wilkie conjecture is known to fail for general o-minimal structures, for instance, in ${\mathbb{R}}_{\text{an}}$${\mathbb{R}}_{\text{an }}$R_("an ")\mathbb{R}_{\text {an }}${\mathbb{R}}_{\text{an }}$ [40]. To achieve such asymptotics, it seems one would have to use auxiliary polynomials of degrees $d=(\log H{)}^q$$d=(\log ⁡H{)}^q$d=(log H)^(q)d=(\log H)^{q}$d=(\log ⁡H{)}^q$, and o-minimality places no restrictions on the geometric complexity as a function of $d$$d$dd$d$.

In formulating his conjecture, Wilkie was probably influenced by Khovanskii's theory of fewnomials [25]. The latter implies fairly sharp bounds for the number of connected components of sets defined using algebraic and exponential functions (and more generally Pfaffian functions) as a function of the degrees of the equations involved. Below we attempt to axiomatize what it would mean for an arbitrary o-minimal structure to satisfy such sharp complexity bounds.

In this section we introduce sharply o-minimal structures, which are meant to endow a standard o-minimal structure with an appropriate notions comparable to dimension and degree in the algebraic case, and provide suitable control over these parameters under the basic logical operations. We first introduce the notion of a format-degree filtration (abbreviated $FD$$FD$FDF D$FD$-filtration) on a structure $S$$S$SS$S$. This is a collection $\mathrm{\Omega }={\left\{{\mathrm{\Omega }}_{\mathcal{F},D}\right\}}_{\mathcal{F},D\in \mathbb{N}}$$\mathrm{Î\copyright }={\left\{{\mathrm{Î\copyright }}_{\mathcal{F},D}\right\}}_{\mathcal{F},D∈\mathbb{N}}$Omega={Omega_(F,D)}_(F,D inN)\Omega=\left\{\Omega_{\mathcal{F}, D}\right\}_{\mathcal{F}, D \in \mathbb{N}}$\mathrm{Î\copyright }={\left\{{\mathrm{Î\copyright }}_{\mathcal{F},D}\right\}}_{\mathcal{F},D∈\mathbb{N}}$ such that each
${\mathrm{\Omega }}_{\mathcal{F},D}$${\mathrm{Î\copyright }}_{\mathcal{F},D}$Omega_(F,D)\Omega_{\mathscr{F}, D}${\mathrm{Î\copyright }}_{\mathcal{F},D}$ is a collection of definable sets (possibly of different ambient dimensions), with

and $\bigcup _{\mathcal{F},D}\mathrm{\Omega }$$\underset{\mathcal{F},D}{⋃} \mathrm{Î\copyright }$uuu_(F,D)Omega\bigcup_{\mathcal{F}, D} \Omega$\underset{\mathcal{F},D}{⋃}\mathrm{Î\copyright }$ is the collection of all definable sets in $\mathcal{S}$$\mathcal{S}$S\mathcal{S}$\mathcal{S}$. We call the sets in ${\mathrm{\Omega }}_{\mathcal{F},D}$${\mathrm{Î\copyright }}_{\mathcal{F},D}$Omega_(F,D)\Omega_{\mathcal{F}, D}${\mathrm{Î\copyright }}_{\mathcal{F},D}$ sets of format $\mathcal{F}$$\mathcal{F}$F\mathscr{F}$\mathcal{F}$ and degree D. However, note that the format and degree of a set are not uniquely defined since $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ is a filtration rather than a partition.

We now come to the notion of a sharply o-minimal structure.

Definition 3 (Sharply o-minimal structure). A sharply o-minimal structure is a pair $\mathrm{\Sigma }:=(S,\mathrm{\Omega })$$\mathrm{Î\pounds }:=(S,\mathrm{Î\copyright })$Sigma:=(S,Omega)\Sigma:=(S, \Omega)$\mathrm{Î\pounds }:=(S,\mathrm{Î\copyright })$ consisting of an o-minimal expansion of the real field $S$$S$SS$S$ and an FD-filtration $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$; and for each $\mathcal{F}\in \mathbb{N}$$\mathcal{F}∈\mathbb{N}$FinN\mathscr{F} \in \mathbb{N}$\mathcal{F}∈\mathbb{N}$, a polynomial $P_{\mathcal{F}}(\cdot )$$P_{\mathcal{F}}(â‹\dots )$P_(F)(*)P_{\mathcal{F}}(\cdot)$P_{\mathcal{F}}(â‹\dots )$ such that the following holds:

If $A\in {\mathrm{\Omega }}_{\mathcal{F},D}$$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$A inOmega_(F,D)A \in \Omega_{\mathcal{F}, D}$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$ then

(1) if $A\subset \mathbb{R}$$A⊂\mathbb{R}$A subRA \subset \mathbb{R}$A⊂\mathbb{R}$, it has at most $P_{\mathcal{F}}(D)$$P_{\mathcal{F}}(D)$P_(F)(D)P_{\mathcal{F}}(D)$P_{\mathcal{F}}(D)$ connected components,

(2) if $A\subset {\mathbb{R}}^{\ell }$$A⊂{\mathbb{R}}^{\mathrm{â„“}}$A subR^(â„“)A \subset \mathbb{R}^{\ell}$A⊂{\mathbb{R}}^{\mathrm{â„“}}$ then $\mathcal{F}⩾\ell$$\mathcal{F}â\copyright ¾\mathrm{â„“}$F >= â„“\mathcal{F} \geqslant \ell$\mathcal{F}â\copyright ¾\mathrm{â„“}$,

(3) $A^c,{\pi }_{\ell -1}(A),A\times \mathbb{R}$$A^c,{\mathrm{Ï€}}_{\mathrm{â„“}−1}(A),A×\mathbb{R}$A^(c),pi_(â„“-1)(A),A xxRA^{c}, \pi_{\ell-1}(A), A \times \mathbb{R}$A^c,{\mathrm{Ï€}}_{\mathrm{â„“}−1}(A),A×\mathbb{R}$, and $\mathbb{R}\times A$$\mathbb{R}×A$Rxx A\mathbb{R} \times A$\mathbb{R}×A$ lie in ${\mathrm{\Omega }}_{\mathcal{F}+1,D}$${\mathrm{Î\copyright }}_{\mathcal{F}+1,D}$Omega_(F+1,D)\Omega_{\mathscr{F}+1, D}${\mathrm{Î\copyright }}_{\mathcal{F}+1,D}$.

Similarly if $A_1,\dots ,A_k\subset {\mathbb{R}}^{\ell }$$A_1,…,A_k⊂{\mathbb{R}}^{\mathrm{â„“}}$A_(1),dots,A_(k)subR^(â„“)A_{1}, \ldots, A_{k} \subset \mathbb{R}^{\ell}$A_1,…,A_k⊂{\mathbb{R}}^{\mathrm{â„“}}$ with $A_j\in {\mathrm{\Omega }}_{{\mathcal{F}}_j,D_j}$$A_j∈{\mathrm{Î\copyright }}_{{\mathcal{F}}_j,D_j}$A_(j)inOmega_(F_(j),D_(j))A_{j} \in \Omega_{\mathcal{F}_{j}, D_{j}}$A_j∈{\mathrm{Î\copyright }}_{{\mathcal{F}}_j,D_j}$ then

where $\mathcal{F}:=\underset{j}{max}{\mathcal{F}}_j$$\mathcal{F}:=\underset{j}{max} {\mathcal{F}}_j$F:=max_(j)F_(j)\mathscr{F}:=\max _{j} \mathscr{F}_{j}$\mathcal{F}:=\underset{j}{max}{\mathcal{F}}_j$ and $D=\sum _j{D}_j$$D=\underset{j}{∑} D_j$D=sum_(j)D_(j)D=\sum_{j} D_{j}$D=\underset{j}{∑}{D}_j$. Finally,

(6) if $P\in \mathbb{R}[x_1,\dots ,x_{\ell }]$$P∈\mathbb{R}\left[x_1,…,x_{\mathrm{â„“}}\right]$P inR[x_(1),dots,x_(â„“)]P \in \mathbb{R}\left[x_{1}, \ldots, x_{\ell}\right]$P∈\mathbb{R}[x_1,…,x_{\mathrm{â„“}}]$ then $\{P=0\}\in {\mathrm{\Omega }}_{\ell ,\mathrm{deg}}P$$\{P=0\}∈{\mathrm{Î\copyright }}_{\mathrm{â„“},\mathrm{deg}}P${P=0}inOmega_(â„“,deg)P\{P=0\} \in \Omega_{\ell, \operatorname{deg}} P$\{P=0\}∈{\mathrm{Î\copyright }}_{\mathrm{â„“},\mathrm{deg}}P$.

Given a collection $\left\{A_{\alpha }\right\}$$\left\{A_{\mathrm{Î\pm }}\right\}${A_(alpha)}\left\{A_{\alpha}\right\}$\left\{A_{\mathrm{Î\pm }}\right\}$ of sets generating a structure $\mathcal{S}$$\mathcal{S}$S\mathcal{S}$\mathcal{S}$, and associated formats and degrees ${\mathcal{F}}_{\alpha },D_{\alpha }$${\mathcal{F}}_{\mathrm{Î\pm }},D_{\mathrm{Î\pm }}$F_(alpha),D_(alpha)\mathcal{F}_{\alpha}, D_{\alpha}${\mathcal{F}}_{\mathrm{Î\pm }},D_{\mathrm{Î\pm }}$ one can consider the minimal FD-filtration $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ satisfying the axioms (2)-(6) above. We call this the FD-filtration generated by $\left\{(A_{\alpha },{\mathcal{F}}_{\alpha },D_{\alpha })\right\}$$\left\{\left(A_{\mathrm{Î\pm }},{\mathcal{F}}_{\mathrm{Î\pm }},D_{\mathrm{Î\pm }}\right)\right\}${(A_(alpha),F_(alpha),D_(alpha))}\left\{\left(A_{\alpha}, \mathcal{F}_{\alpha}, D_{\alpha}\right)\right\}$\left\{(A_{\mathrm{Î\pm }},{\mathcal{F}}_{\mathrm{Î\pm }},D_{\mathrm{Î\pm }})\right\}$. This will be sharply ominimal if and only if axiom (1) is satisfied.

Definition 4 (Reduction of FD-filtrations). Let $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright },{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega,Omega^(')\Omega, \Omega^{\prime}$\mathrm{Î\copyright },{\mathrm{Î\copyright }}^{\mathrm{′}}$ be two FD-filtrations on a structure $S$$S$SS$S$. We say that $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ is reducible to ${\mathrm{\Omega }}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}^{\mathrm{′}}$Omega^(')\Omega^{\prime}${\mathrm{Î\copyright }}^{\mathrm{′}}$ and write $\mathrm{\Omega }⩽{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega <= Omega^(')\Omega \leqslant \Omega^{\prime}$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$ if there exist functions $a:\mathbb{N}\to \mathbb{N}$$a:\mathbb{N}→\mathbb{N}$a:NrarrNa: \mathbb{N} \rightarrow \mathbb{N}$a:\mathbb{N}→\mathbb{N}$ and $b:\mathbb{N}\to \mathbb{N}[D]$$b:\mathbb{N}→\mathbb{N}[D]$b:NrarrN[D]b: \mathbb{N} \rightarrow \mathbb{N}[D]$b:\mathbb{N}→\mathbb{N}[D]$ such that

We say that $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright },{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega,Omega^(')\Omega, \Omega^{\prime}$\mathrm{Î\copyright },{\mathrm{Î\copyright }}^{\mathrm{′}}$ are equivalent if $\mathrm{\Omega }⩽{\mathrm{\Omega }}^{\mathrm{\prime }}⩽\mathrm{\Omega }$$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}â\copyright ½\mathrm{Î\copyright }$Omega <= Omega^(') <= Omega\Omega \leqslant \Omega^{\prime} \leqslant \Omega$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}â\copyright ½\mathrm{Î\copyright }$.

We will usually try to prove that certain measures of complexity of definable sets depend polynomially on the degree, thinking of the format as constant. If one can prove such a statement for ${\mathrm{\Omega }}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}^{\mathrm{′}}$Omega^(')\Omega^{\prime}${\mathrm{Î\copyright }}^{\mathrm{′}}$-degrees, and $\mathrm{\Omega }⩽{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega <= Omega^(')\Omega \leqslant \Omega^{\prime}$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$, then the same statement holds for $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$-degrees and in this sense $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ is reducible to ${\mathrm{\Omega }}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}^{\mathrm{′}}$Omega^(')\Omega^{\prime}${\mathrm{Î\copyright }}^{\mathrm{′}}$.

Remark 5 (Effectivity). One can require further that a sharply o-minimal structure is effective, in the sense that the polynomial $P_{\mathcal{F}}(D)$$P_{\mathcal{F}}(D)$P_(F)(D)P_{\mathcal{F}}(D)$P_{\mathcal{F}}(D)$ in Definition 3 is given by some explicit primitive recursive function of $\mathcal{F}$$\mathcal{F}$F\mathscr{F}$\mathcal{F}$. Similarly, one may require a reduction $\mathrm{\Omega }⩽{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega <= Omega^(')\Omega \leqslant \Omega^{\prime}$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$ to be effective. Unless otherwise stated, all constructions in this paper are effective in this sense.

Consider the structure ${\mathbb{R}}_{\text{alg}}$${\mathbb{R}}_{\text{alg }}$R_("alg ")\mathbb{R}_{\text {alg }}${\mathbb{R}}_{\text{alg }}$ with the FD-filtration $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ generated by all algebraic hypersurfaces $\{P=0\}$$\{P=0\}${P=0}\{P=0\}$\{P=0\}$ with the format given by the ambient dimension and the degree given by $\mathrm{deg}P$$\mathrm{deg}⁡P$deg P\operatorname{deg} P$\mathrm{deg}⁡P$. Then $({\mathbb{R}}_{\text{alg}},\mathrm{\Omega })$$\left({\mathbb{R}}_{\text{alg }},\mathrm{Î\copyright }\right)$(R_("alg "),Omega)\left(\mathbb{R}_{\text {alg }}, \Omega\right)$({\mathbb{R}}_{\text{alg }},\mathrm{Î\copyright })$ is a sharply o-minimal structure. This is not an immediate statement: it follows from the results on effective cell decomposition, or elimination of quantifiers, in semialgebraic geometry $[3]$$[3]$[3][3]$[3]$.

Perhaps a more natural notion of format and degree in the semialgebraic category is as follows. Define ${\mathrm{\Omega }}_{\mathcal{F},D}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}_{\mathcal{F},D}^{\mathrm{′}}$Omega_(F,D)^(')\Omega_{\mathcal{F}, D}^{\prime}${\mathrm{Î\copyright }}_{\mathcal{F},D}^{\mathrm{′}}$ to be the subsets of ${\mathbb{R}}^{\ell }$${\mathbb{R}}^{\mathrm{â„“}}$R^(â„“)\mathbb{R}^{\ell}${\mathbb{R}}^{\mathrm{â„“}}$ with $\ell ⩽\mathcal{F}$$\mathrm{â„“}â\copyright ½\mathcal{F}$â„“ <= F\ell \leqslant \mathcal{F}$\mathrm{â„“}â\copyright ½\mathcal{F}$, that can be written as a union of basic sets

with the sum of the degrees of the $P_i$$P_i$P_(i)P_{i}$P_i$ and $Q_j$$Q_j$Q_(j)Q_{j}$Q_j$, over all basic sets, bounded by $D$$D$DD$D$. This is not sharply o-minimal according to our definition because it does not satisfy axiom (3), for instance. However, it is equivalent to $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ defined above.

Not surprisingly, ${\mathbb{R}}_{\text{an}}$${\mathbb{R}}_{\text{an }}$R_("an ")\mathbb{R}_{\text {an }}${\mathbb{R}}_{\text{an }}$ is not sharply o-minimal with respect to any FD-filtration. Assume the contrary. Let ${\omega }_1=1$${\mathrm{ω}}_1=1$omega_(1)=1\omega_{1}=1${\mathrm{ω}}_1=1$ and ${\omega }_{n+1}=2^{{\omega }_n}$${\mathrm{ω}}_{n+1}=2^{{\mathrm{ω}}_n}$omega_(n+1)=2^(omega_(n))\omega_{n+1}=2^{\omega_{n}}${\mathrm{ω}}_{n+1}=2^{{\mathrm{ω}}_n}$, and let $\mathrm{\Gamma }=\{y=f(z)\}\subset {\mathbb{C}}^2$$\mathrm{Γ}=\{y=f(z)\}⊂{\mathbb{C}}^2$Gamma={y=f(z)}subC^(2)\Gamma=\{y=f(z)\} \subset \mathbb{C}^{2}$\mathrm{Γ}=\{y=f(z)\}⊂{\mathbb{C}}^2$ denote the graph of the holomorphic function $f(z)=\sum _{j=1}^{\mathrm{\infty }}{z}^{{\omega }_j}$$f(z)=\underset{j=1}{\overset{\mathrm{∞}}{∑}} z^{{\mathrm{ω}}_j}$f(z)=sum_(j=1)^(oo)z^(omega_(j))f(z)=\sum_{j=1}^{\infty} z^{\omega_{j}}$f(z)=\underset{j=1}{\overset{\mathrm{∞}}{∑}}{z}^{{\mathrm{ω}}_j}$ restricted to the disc of radius $1/2$$1/2$1//21 / 2$1/2$ (which is definable in ${\mathbb{R}}_{\mathrm{a}\mathrm{n}}$${\mathbb{R}}_{\mathrm{a}\mathrm{n}}$R_(an)\mathbb{R}_{\mathrm{an}}${\mathbb{R}}_{\mathrm{a}\mathrm{n}}$ ). Then by axioms (1), (5) and (6), the number of points in

should be polynomial in ${\omega }_n$${\mathrm{ω}}_n$omega_(n)\omega_{n}${\mathrm{ω}}_n$, with the exact polynomial depending on the format and degree of $\mathrm{\Gamma }$$\mathrm{Γ}$Gamma\Gamma$\mathrm{Γ}$. But it is, in fact, ${\omega }_{n+1}=2^{{\omega }_n}$${\mathrm{ω}}_{n+1}=2^{{\mathrm{ω}}_n}$omega_(n+1)=2^(omega_(n))\omega_{n+1}=2^{\omega_{n}}${\mathrm{ω}}_{n+1}=2^{{\mathrm{ω}}_n}$ for $0<\epsilon \ll 1$$0<\mathrm{Î\mu }≪1$0 < epsi≪10<\varepsilon \ll 1$0<\mathrm{Î\mu }≪1$, and we have a contradiction for $n\gg 1$$n≫1$n≫1n \gg 1$n≫1$.

Let $B\subset {\mathbb{R}}^{\ell }$$B⊂{\mathbb{R}}^{\mathrm{ℓ}}$B subR^(ℓ)B \subset \mathbb{R}^{\ell}$B⊂{\mathbb{R}}^{\mathrm{ℓ}}$ be a domain, which for simplicity we take to be a product of (possibly infinite) intervals. A tuple $f_1,\dots ,f_m:B\to \mathbb{R}$$f_1,…,f_m:B→\mathbb{R}$f_(1),dots,f_(m):B rarrRf_{1}, \ldots, f_{m}: B \rightarrow \mathbb{R}$f_1,…,f_m:B→\mathbb{R}$ of analytic functions is called a Pfaffian chain if they satisfy a triangular system of algebraic differential equations of the form

They are called restricted if $B$$B$BB$B$ is bounded and $f_1,\dots ,f_m$$f_1,…,f_m$f_(1),dots,f_(m)f_{1}, \ldots, f_{m}$f_1,…,f_m$ extend as real analytic functions to $\overline{B}$$\stackrel{¯}{B}$bar(B)\bar{B}$\stackrel{¯}{B}$. A Pfaffian function is a polynomial $Q(x_1,\dots ,x_{\ell },f_1,\dots ,f_m)$$Q\left(x_1,…,x_{\mathrm{ℓ}},f_1,…,f_m\right)$Q(x_(1),dots,x_(ℓ),f_(1),dots,f_(m))Q\left(x_{1}, \ldots, x_{\ell}, f_{1}, \ldots, f_{m}\right)$Q(x_1,…,x_{\mathrm{ℓ}},f_1,…,f_m)$. We denote the structure generated by the Pfaffian functions by ${\mathbb{R}}_{\text{Pfaff}}$${\mathbb{R}}_{\text{Pfaff }}$R_("Pfaff ")\mathbb{R}_{\text {Pfaff }}${\mathbb{R}}_{\text{Pfaff }}$, and its restricted analog by ${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$R_(rPfaff)\mathbb{R}_{\mathrm{rPfaff}}${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$.

Khovanskii [25] proved upper bounds for the number of connected components of systems of Pfaffian equations. This was later extended by Gabrielov and Vorobjov to sets defined using inequalities and quantifiers [21]. However, their results fall short of establishing the sharp o-minimality of ${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$R_(rPffaff)\mathbb{R}_{\mathrm{rPffaff}}${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$. The problem is that for Gabrielov-Vorobjov's notion of format and degree, if $A\in {\mathrm{\Omega }}_{\mathcal{F},D}$$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$A inOmega_(F,D)A \in \Omega_{\mathscr{F}, D}$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$ then they are only able to show that $A^c\in {\mathrm{\Omega }}_{P_{\mathcal{F}}(D),P_{\mathcal{F}}(D)}$$A^c∈{\mathrm{Î\copyright }}_{P_{\mathcal{F}}(D),P_{\mathcal{F}}(D)}$A^(c)inOmega_(P_(F)(D),P_(F)(D))A^{c} \in \Omega_{P_{\mathscr{F}}(D), P_{\mathscr{F}}(D)}$A^c∈{\mathrm{Î\copyright }}_{P_{\mathcal{F}}(D),P_{\mathcal{F}}(D)}$
rather than $A^c\in {\mathrm{\Omega }}_{\mathcal{F}+1,P_{\mathcal{F}}(D)}$$A^c∈{\mathrm{Î\copyright }}_{\mathcal{F}+1,P_{\mathcal{F}}(D)}$A^(c)inOmega_(F+1,P_(F)(D))A^{c} \in \Omega_{\mathscr{F}+1, P_{\mathscr{F}}(D)}$A^c∈{\mathrm{Î\copyright }}_{\mathcal{F}+1,P_{\mathcal{F}}(D)}$ as required by our axioms. This is a fundamental difficulty, as it is essential in our setup that the format never becomes dependent on the degree.

In [13] the first author and Vorobjov introduce a modified notion of format and degree and prove the following.

Theorem 6. There is an $FD$$FD$FDF D$FD$-filtration $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$ on ${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$R_(rPfaff)\mathbb{R}_{\mathrm{rPfaff}}${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$ that makes it into a sharply o-minimal structure. Moreover, Gabrielov-Vorobjov's standard filtration is reducible to $\mathrm{\Omega }$$\mathrm{Î\copyright }$Omega\Omega$\mathrm{Î\copyright }$.

We conjecture that this theorem extends to the structure ${\mathbb{R}}_{\text{Pfaff}}$${\mathbb{R}}_{\text{Pfaff }}$R_("Pfaff ")\mathbb{R}_{\text {Pfaff }}${\mathbb{R}}_{\text{Pfaff }}$, and this is the subject of work in progress by the first author and Vorobjov utilizing some additional ideas of Gabrielov [19].

We recall the notion of a cell in an o-minimal structure. A cell $C\subset \mathbb{R}$$C⊂\mathbb{R}$C subRC \subset \mathbb{R}$C⊂\mathbb{R}$ is either a point or an open interval (possibly infinite). A cell $C\subset {\mathbb{R}}^{\ell +1}$$C⊂{\mathbb{R}}^{\mathrm{ℓ}+1}$C subR^(ℓ+1)C \subset \mathbb{R}^{\ell+1}$C⊂{\mathbb{R}}^{\mathrm{ℓ}+1}$ is either the graph of a definable continuous function $f:C^{\mathrm{\prime }}\to \mathbb{R}$$f:C^{\mathrm{′}}→\mathbb{R}$f:C^(')rarrRf: C^{\prime} \rightarrow \mathbb{R}$f:C^{\mathrm{′}}→\mathbb{R}$ where $C^{\mathrm{\prime }}\subset {\mathbb{R}}^{\ell }$$C^{\mathrm{′}}⊂{\mathbb{R}}^{\mathrm{ℓ}}$C^(')subR^(ℓ)C^{\prime} \subset \mathbb{R}^{\ell}$C^{\mathrm{′}}⊂{\mathbb{R}}^{\mathrm{ℓ}}$ is a cell, or the area strictly between two graphs of such definable continuous functions $f,g:C^{\mathrm{\prime }}\to \mathbb{R}$$f,g:C^{\mathrm{′}}→\mathbb{R}$f,g:C^(')rarrRf, g: C^{\prime} \rightarrow \mathbb{R}$f,g:C^{\mathrm{′}}→\mathbb{R}$ satisfying $f<g$$f<g$f < gf<g$f<g$ identically on $C^{\mathrm{\prime }}$$C^{\mathrm{′}}$C^(')C^{\prime}$C^{\mathrm{′}}$. One can also take $f=-\mathrm{\infty }$$f=−\mathrm{∞}$f=-oof=-\infty$f=−\mathrm{∞}$ and $g=\mathrm{\infty }$$g=\mathrm{∞}$g=oog=\infty$g=\mathrm{∞}$ in this definition.

We say that a cell $C\subset {\mathbb{R}}^{\ell }$$C⊂{\mathbb{R}}^{\mathrm{ℓ}}$C subR^(ℓ)C \subset \mathbb{R}^{\ell}$C⊂{\mathbb{R}}^{\mathrm{ℓ}}$ is compatible with $X\subset {\mathbb{R}}^{\ell }$$X⊂{\mathbb{R}}^{\mathrm{ℓ}}$X subR^(ℓ)X \subset \mathbb{R}^{\ell}$X⊂{\mathbb{R}}^{\mathrm{ℓ}}$ if it is either strictly contained, or strictly disjoint from $X$$X$XX$X$. The following cell decomposition theorem can be viewed as the raison d'être of the axioms of o-minimality.

Theorem 7 (Cell decomposition). Let $X_1,\dots ,X_k\subset {\mathbb{R}}^{\ell }$$X_1,…,X_k⊂{\mathbb{R}}^{\mathrm{ℓ}}$X_(1),dots,X_(k)subR^(ℓ)X_{1}, \ldots, X_{k} \subset \mathbb{R}^{\ell}$X_1,…,X_k⊂{\mathbb{R}}^{\mathrm{ℓ}}$ be definable sets. Then there is a decomposition of ${\mathbb{R}}^{\ell }$${\mathbb{R}}^{\mathrm{ℓ}}$R^(ℓ)\mathbb{R}^{\ell}${\mathbb{R}}^{\mathrm{ℓ}}$ into pairwise disjoint cells that are pairwise compatible with $X_1,\dots ,X_k$$X_1,…,X_k$X_(1),dots,X_(k)X_{1}, \ldots, X_{k}$X_1,…,X_k$.

Given the importance of cell decomposition in the theory of o-minimality, it is natural to pose the following question.

Question 8. If $S$$S$SS$S$ is sharply o-minimal and $X_1,\dots ,X_k$$X_1,…,X_k$X_(1),dots,X_(k)X_{1}, \ldots, X_{k}$X_1,…,X_k$ have format $\mathcal{F}$$\mathcal{F}$F\mathscr{F}$\mathcal{F}$ and degree $D$$D$DD$D$, can one find a cell-decomposition where each cell has format $\mathrm{const}(\mathcal{F})$$\mathrm{const}⁡(\mathcal{F})$const(F)\operatorname{const}(\mathcal{F})$\mathrm{const}⁡(\mathcal{F})$, and the number of cells and their degrees are bounded by poly ${}_{\mathcal{F},\mathcal{F}}(k,D)$${}_{\mathcal{F},\mathcal{F}}(k,D)$_(F,F)(k,D){ }_{\mathcal{F}, \mathcal{F}}(k, D)${}_{\mathcal{F},\mathcal{F}}(k,D)$ ?

We suspect the answer to this question may be negative. Since cell decomposition is perhaps the most crucial construction in o-minimality, this is a fundamental problem. The following result rectifies the situation.

Theorem 9. Let $(\mathcal{S},\mathrm{\Omega })$$(\mathcal{S},\mathrm{Î\copyright })$(S,Omega)(\mathcal{S}, \Omega)$(\mathcal{S},\mathrm{Î\copyright })$ be sharply o-minimal. Then there exists another $FD$$FD$FDF D$FD$-filtration ${\mathrm{\Omega }}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}^{\mathrm{′}}$Omega^(')\Omega^{\prime}${\mathrm{Î\copyright }}^{\mathrm{′}}$ with ( $S,{\mathrm{\Omega }}^{\mathrm{\prime }}$$S,{\mathrm{Î\copyright }}^{\mathrm{′}}$S,Omega^(')S, \Omega^{\prime}$S,{\mathrm{Î\copyright }}^{\mathrm{′}}$ ) sharply o-minimal such that $\mathrm{\Omega }⩽{\mathrm{\Omega }}^{\mathrm{\prime }}$$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$Omega <= Omega^(')\Omega \leqslant \Omega^{\prime}$\mathrm{Î\copyright }â\copyright ½{\mathrm{Î\copyright }}^{\mathrm{′}}$, and in ${\mathrm{\Omega }}^{\mathrm{\prime }}$${\mathrm{Î\copyright }}^{\mathrm{′}}$Omega^(')\Omega^{\prime}${\mathrm{Î\copyright }}^{\mathrm{′}}$ the following holds.

Let $X_1,\dots ,X_k\in {\mathrm{\Omega }}_{\mathcal{F},D}^{\mathrm{\prime }}$$X_1,…,X_k∈{\mathrm{Î\copyright }}_{\mathcal{F},D}^{\mathrm{′}}$X_(1),dots,X_(k)inOmega_(F,D)^(')X_{1}, \ldots, X_{k} \in \Omega_{\mathcal{F}, D}^{\prime}$X_1,…,X_k∈{\mathrm{Î\copyright }}_{\mathcal{F},D}^{\mathrm{′}}$, all subsets of ${\mathbb{R}}^{\ell }$${\mathbb{R}}^{\mathrm{â„“}}$R^(â„“)\mathbb{R}^{\ell}${\mathbb{R}}^{\mathrm{â„“}}$. Then there exists a cell decomposition of ${\mathbb{R}}^{\ell }$${\mathbb{R}}^{\mathrm{â„“}}$R^(â„“)\mathbb{R}^{\ell}${\mathbb{R}}^{\mathrm{â„“}}$ compatible with each $X_j$$X_j$X_(j)X_{j}$X_j$ such that each cell has format const $(\mathcal{F})$$(\mathcal{F})$(F)(\mathscr{F})$(\mathcal{F})$, the number of cells is poly $\mathcal{F}(k,D)$$\mathcal{F}(k,D)$F(k,D)\mathcal{F}(k, D)$\mathcal{F}(k,D)$, and the degree of each cell is ${\mathrm{poly}}_{\mathcal{F}}(D)$${\mathrm{poly}}_{\mathcal{F}}⁡(D)$poly_(F)(D)\operatorname{poly}_{\mathcal{F}}(D)${\mathrm{poly}}_{\mathcal{F}}⁡(D)$.

In the structure ${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$R_(rPfaff)\mathbb{R}_{\mathrm{rPfaff}}${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$, Theorem 9 is one of the main results of [13]. The general case is obtained by generalizing the proof to the general sharply o-minimal case, and is part of the $\mathrm{P}\mathrm{h}\mathrm{D}$$\mathrm{P}\mathrm{h}\mathrm{D}$PhD\mathrm{PhD}$\mathrm{P}\mathrm{h}\mathrm{D}$ thesis of Binyamin Zack-Kutuzov.

Let $I:=(0,1)$$I:=(0,1)$I:=(0,1)I:=(0,1)$I:=(0,1)$. For $f:I^n\to {\mathbb{R}}^m$$f:I^n→{\mathbb{R}}^m$f:I^(n)rarrR^(m)f: I^{n} \rightarrow \mathbb{R}^{m}$f:I^n→{\mathbb{R}}^m$ a $C^r$$C^r$C^(r)C^{r}$C^r$-smooth map, we denote

The Yomdin-Gromov algebraic lemma is a result about $C^r$$C^r$C^(r)C^{r}$C^r$-smooth parametrizations of bounded norm for definable subsets of $I^n$$I^n$I^(n)I^{n}$I^n$. A sharply o-minimal version of this lemma is as follows.

Lemma 10. Let $(\mathcal{S},\mathrm{\Omega })$$(\mathcal{S},\mathrm{Î\copyright })$(S,Omega)(\mathcal{S}, \Omega)$(\mathcal{S},\mathrm{Î\copyright })$ be sharply o-minimal. Then there is a polynomial $P_{\mathcal{F},r}(\cdot )$$P_{\mathcal{F},r}(â‹\dots )$P_(F,r)(*)P_{\mathcal{F}, r}(\cdot)$P_{\mathcal{F},r}(â‹\dots )$ depending on the pair $(\mathcal{F},r)$$(\mathcal{F},r)$(F,r)(\mathcal{F}, r)$(\mathcal{F},r)$, such that for every $A\in {\mathrm{\Omega }}_{\mathcal{F},D}$$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$A inOmega_(F,D)A \in \Omega_{\mathcal{F}, D}$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$ the following holds. There exist a collection of maps $\{f_{\alpha }:I^{n_{\alpha }}\to A\}$$\left\{f_{\mathrm{Î\pm }}:I^{n_{\mathrm{Î\pm }}}→A\right\}${f_(alpha):I^(n_(alpha))rarr A}\left\{f_{\alpha}: I^{n_{\alpha}} \rightarrow A\right\}$\{f_{\mathrm{Î\pm }}:I^{n_{\mathrm{Î\pm }}}→A\}$ of size at most $P_{\mathcal{F},r}(D)$$P_{\mathcal{F},r}(D)$P_(F,r)(D)P_{\mathcal{F}, r}(D)$P_{\mathcal{F},r}(D)$ such that $\bigcup _{\alpha }{f}_{\alpha }\left(I^{n_{\alpha }}\right)=A$$\underset{\mathrm{Î\pm }}{⋃} f_{\mathrm{Î\pm }}\left(I^{n_{\mathrm{Î\pm }}}\right)=A$uuu_(alpha)f_(alpha)(I^(n_(alpha)))=A\bigcup_{\alpha} f_{\alpha}\left(I^{n_{\alpha}}\right)=A$\underset{\mathrm{Î\pm }}{⋃}{f}_{\mathrm{Î\pm }}\left(I^{n_{\mathrm{Î\pm }}}\right)=A$; and ${\parallel f_{\alpha }\parallel }_r⩽1$${∥f_{\mathrm{Î\pm }}∥}_râ\copyright ½1$||f_(alpha)||_(r) <= 1\left\|f_{\alpha}\right\|_{r} \leqslant 1${âˆ\yen f_{\mathrm{Î\pm }}âˆ\yen }_râ\copyright ½1$ and $n_{\alpha }⩽\dim A$$n_{\mathrm{Î\pm }}â\copyright ½\dim ⁡A$n_(alpha) <= dim An_{\alpha} \leqslant \operatorname{dim} A$n_{\mathrm{Î\pm }}â\copyright ½\dim ⁡A$ for every $\alpha$$\mathrm{Î\pm }$alpha\alpha$\mathrm{Î\pm }$.

In the algebraic case, this result is due to Gromov [23], based on a similar but slightly more technically involved statement by Yomdin [44]. In the general o-minimal case, but without complexity bounds, the result is due to Pila and Wilkie [37]. In the restricted Pfaffian case, this result is due to the first author with Jones, Schmidt, and Thomas [6] using Theorem 9 in the ${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$R_(rPfaff)\mathbb{R}_{\mathrm{rPfaff}}${\mathbb{R}}_{\mathrm{r}\mathrm{P}\mathrm{f}\mathrm{a}\mathrm{f}\mathrm{f}}$ case. The general case follows in the same way.

The following conjecture seems plausible, though we presently do not have an approach to proving it in this generality.

Conjecture 11. In Lemma 10, one can replace $P_{\mathcal{F},r}(D)$$P_{\mathcal{F},r}(D)$P_(F,r)(D)P_{\mathcal{F}, r}(D)$P_{\mathcal{F},r}(D)$ by a $P_{\mathcal{F}}(D,r)$$P_{\mathcal{F}}(D,r)$P_(F)(D,r)P_{\mathcal{F}}(D, r)$P_{\mathcal{F}}(D,r)$, i.e., by a polynomial in both $D$$D$DD$D$ and $r$$r$rr$r$, depending only on $\mathcal{F}$$\mathcal{F}$F\mathcal{F}$\mathcal{F}$.

In the structure ${\mathbb{R}}_{\text{alg}}$${\mathbb{R}}_{\text{alg }}$R_("alg ")\mathbb{R}_{\text {alg }}${\mathbb{R}}_{\text{alg }}$ this was conjectured by Yomdin (unpublished) and by Burguet [14], in relation to a conjecture of Yomdin [45, CONJECTURE 6.1] concerning the rate of decay of the tail entropy for real-analytic mappings. The conjecture was proved in [9] by complexanalytic methods. We will say more about the possible generalization of these methods to more general sharply o-minimal structures in Section 3.

We now state a form of the Pila-Wilkie counting theorem, Theorem 1, with explicit control over the asymptotic constant.

Theorem 12. Let $(S,\mathrm{\Omega })$$(S,\mathrm{Î\copyright })$(S,Omega)(S, \Omega)$(S,\mathrm{Î\copyright })$ be sharply o-minimal. Then for every $ϵ>0$$\mathrm{Ï\mu }>0$epsilon > 0\epsilon>0$\mathrm{Ï\mu }>0$ and $\mathcal{F}$$\mathcal{F}$F\mathcal{F}$\mathcal{F}$ there is a polynomial $P_{\mathcal{F},ϵ}(\cdot )$$P_{\mathcal{F},\mathrm{Ï\mu }}(â‹\dots )$P_(F,epsilon)(*)P_{\mathcal{F}, \epsilon}(\cdot)$P_{\mathcal{F},\mathrm{Ï\mu }}(â‹\dots )$ depending on $(\mathcal{S},\mathrm{\Omega })$$(\mathcal{S},\mathrm{Î\copyright })$(S,Omega)(\mathcal{S}, \Omega)$(\mathcal{S},\mathrm{Î\copyright })$, such that for every $A\in {\mathrm{\Omega }}_{\mathcal{F},D}$$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$A inOmega_(F,D)A \in \Omega_{\mathcal{F}, D}$A∈{\mathrm{Î\copyright }}_{\mathcal{F},D}$ and $H⩾2$$Hâ\copyright ¾2$H >= 2H \geqslant 2$Hâ\copyright ¾2$,